Joseph L. Pe - cp's OEIS frontend

A333079 The largest nontrivial divisor of n equals the sum of the other nontrivial divisors of n.

345, 1645, 6489, 8041, 23881, 88473, 115957, 342637, 3256261, 4114285, 4646101, 5054221, 13384681, 17897737, 20901553, 23807821, 42081409, 64580041, 65380921, 70366153, 82175857, 110344621, 137331565, 164109901, 286078081, 331957897, 366611617, 367891717, 489645157

Offset: 1

Joseph L. Pe, Mar 07 2020

nonn

A divisor of n other than 1 and n is called a nontrivial divisor of n.
In general, if p, p+k, and q = (p^2+(2+k)*p+k)/(k-1) are 3 primes and p < p+k < q, then p(p+k)q is a term. In particular, if p, p+2, and p^2+4*p+2 are 3 primes, then p(p+2)(p^2+4*p+2) is a term. - Giovanni Resta, Mar 08 2020
Each term in this sequence has at least eight divisors. - Bernard Schott, Mar 09 2020

			The nontrivial divisors of 345 are 3, 5, 15, 23, 69, 115, the largest of which, 115, is equal to the sum of the other nontrivial divisors 3, 5, 15, 23, 69.

Cf. A032742.

Select[Range[10^5], 2 # / FactorInteger[#][[1, 1]] == DivisorSigma[1, #] - # - 1 &] (* Giovanni Resta, Mar 07 2020 *)
lndQ[n_]:=With[{c=TakeDrop[Rest[Most[Divisors[n]]],-1]},c[[1,1]]==Total[c[[2]]]]; Select[Range[ 51*10^5],lndQ]//Quiet (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Jan 16 2024 *)

for(k=2,5*10^7,my(d=divisors(k)); if(#d>2&&d[#d-1]==vecsum(d[2..#d-2]), print1(k,", "))) \\ Hugo Pfoertner, Mar 07 2020

More terms from Giovanni Resta, Mar 07 2020

A238161 Greatest common divisor of the prime factors of n, each increased by 1.

Original entry on oeis.org

3, 4, 3, 6, 1, 8, 3, 4, 3, 12, 1, 14, 1, 2, 3, 18, 1, 20, 3, 4, 3, 24, 1, 6, 1, 4, 1, 30, 1, 32, 3, 4, 3, 2, 1, 38, 1, 2, 3, 42, 1, 44, 3, 2, 3, 48, 1, 8, 3, 2, 1, 54, 1, 6, 1, 4, 3, 60, 1, 62, 1, 4, 3, 2, 1, 68, 3, 4, 1, 72, 1, 74, 1, 2, 1, 4, 1, 80, 3, 4, 3, 84, 1, 6, 1, 2, 3, 90, 1, 2, 3, 4, 3, 2, 1, 98, 1, 4, 3

Offset: 2

Joseph L. Pe, Feb 18 2014

			10 has prime factors 2 and 5, which become 3 and 6 when respectively increased by 1, and gcd(3, 6) = 3. Therefore, a(10) = 3.

Table[Apply[GCD, (Transpose[FactorInteger[n]][[1]] + 1)], {n, 2, 100}]

A238162 Least common multiple of the prime factors of n, each increased by 1.

Original entry on oeis.org

3, 4, 3, 6, 12, 8, 3, 4, 6, 12, 12, 14, 24, 12, 3, 18, 12, 20, 6, 8, 12, 24, 12, 6, 42, 4, 24, 30, 12, 32, 3, 12, 18, 24, 12, 38, 60, 28, 6, 42, 24, 44, 12, 12, 24, 48, 12, 8, 6, 36, 42, 54, 12, 12, 24, 20, 30, 60, 12, 62, 96, 8, 3, 42, 12, 68, 18, 24, 24, 72, 12, 74, 114, 12, 60, 24, 84, 80, 6, 4, 42, 84, 24, 18, 132, 60, 12, 90, 12, 56, 24, 32, 48, 60, 12, 98, 24, 12, 6

Offset: 2

Joseph L. Pe, Feb 18 2014

nonn

If n is prime, then a(n) = n + 1. - Wesley Ivan Hurt, Apr 05 2014

If n is a composite squarefree number and a(n) divides n+1, then n is a Lucas-Carmichael number (A006972). - Daniel Suteu, Oct 02 2022

			The prime factors of 6 are 2 and 3, which become 3 and 4 when respectively increased by 1, and lcm(3, 4) = 12. Therefore, a(6) = 12.

Daniel Suteu, Table of n, a(n) for n = 2..10000

Cf. A006972.

a(n) = my(f=factor(n)); lcm(vector(#f~, k, f[k, 1]+1)); \\ Daniel Suteu, Oct 02 2022

A237282 The sum of the totatives of n is a perfect cube.

Original entry on oeis.org

1, 2, 9, 16, 36, 128, 200, 243, 288, 289, 450, 972, 1024, 1156, 1600, 2304, 3600, 6561, 7776, 8192, 8214, 8664, 9126, 9248, 10584, 12150, 12800, 14450, 15987, 18432, 20808, 24843, 25000, 26244, 27075, 28800, 30250, 33075, 51005, 56250, 62208, 63001, 63948

Offset: 1

Joseph L. Pe, Feb 05 2014

nonn

A positive integer <= n that is relatively prime to n is called a totative of n.

			The sum of totatives of 9 is 1 + 2 + 4 + 5 + 7 + 8 = 27 = 3^3; therefore, 9 is a term of the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Totative

Cf. A023896.

g[n_] := Module[{r, i},
    r = {};
    For[i = 1, i <= n, i++,
      If[GCD[n, i] == 1,
        r = Append[r, i]]];
    Apply[Plus, r]];
Select[Range[2*10^3], IntegerQ[g[#]^(1/3)] &]

is(n)=ispower(n*eulerphi(n)/2, 3) || n==1 \\ Charles R Greathouse IV, Sep 21 2016

More terms from Alois P. Heinz, Feb 05 2014

A236693 Numbers k such that 2^sigma(k) == 1 (mod k).

Original entry on oeis.org

1, 3, 15, 35, 51, 65, 105, 119, 195, 255, 315, 323, 357, 377, 455, 459, 585, 595, 663, 969, 1045, 1071, 1105, 1131, 1189, 1365, 1455, 1469, 1485, 1547, 1615, 1785, 1799, 1885, 1887, 1911, 2261, 2295, 2385, 2639, 2795, 2907, 3135, 3145, 3185, 3213, 3315, 3339

Offset: 1

Joseph L. Pe, Jan 30 2014

This sequence is infinite since A051179(n) is a term. - Jinyuan Wang, Mar 13 2020

			2^sigma(15) = 2^24 = 16777216 is congruent to 1 (mod 15), so 15 is a term of the sequence.

Jinyuan Wang, Table of n, a(n) for n = 1..10000
Florian Luca, Positive integers n such that n| a^sigma(n) - 1, Novi Sad Journal of Mathematics, Vol 33, No. 2 (2003), pp. 49-66.

Supersequence of A015715.

Cf. A000203, A051179, A020492, A276238.

l = {1};
For[i = 1, i <= 10^4, i++,
    If[Mod[2^DivisorSigma[1, i], i] == 1, l = Append[l, i]]];
l

s=[1]; for(n=1, 10000, if(2^sigma(n)%n==1, s=concat(s, n))); s \\ Colin Barker, Jan 30 2014

isok(n) = Mod(2, n)^sigma(n)==1; \\ Altug Alkan, Sep 19 2017

a(1) = 1 added by Amiram Eldar, Sep 19 2017

A235989 sigma(n) is an additive inverse of n modulo phi(n).

Original entry on oeis.org

1, 2, 6, 10, 12, 28, 76, 120, 312, 588, 672, 888, 1060, 1264, 1656, 14496, 17900, 22896, 44676, 71712, 77688, 95040, 183600, 233088, 327424, 411264, 425376, 446016, 453258, 655776, 1041120, 1253304, 2708640, 5241856, 5468352, 8676576, 9738912, 12536640, 59489184

Offset: 1

Joseph L. Pe, Jan 27 2014

nonn

sigma(10) = 18 is congruent to 2 = -10 mod 4 and phi(10) = 4; so 10 is a term of the sequence.

If p = 5*2^k-1 is a prime, as it happens for k = 2, 4, 8, 10, 12, 14,... (A001770), then n = 2^k*p is in the sequence, since n+sigma(n) = 6*phi(n). - Giovanni Resta, Jan 27 2014

Giovanni Resta, Table of n, a(n) for n = 1..59 (terms < 10^11)

Cf. A001770.

t = {1}; For[i = 1, i <= 10^6, i++; If[Mod[DivisorSigma[1, i] + i, EulerPhi[i]] == 0, AppendTo[t, i]]]; t

isok(n) = !((sigma(n) + n) % eulerphi(n)); \\ Michel Marcus, Jan 27 2014

More terms from Michel Marcus, Jan 27 2014

A236388 Primes in order of first appearance among the prime factors of p+1 where p is a prime.

Original entry on oeis.org

3, 2, 7, 5, 19, 11, 31, 17, 37, 13, 23, 79, 41, 29, 97, 53, 43, 139, 47, 71, 157, 83, 59, 199, 67, 211, 229, 61, 131, 271, 137, 307, 103, 107, 109, 331, 337, 113, 173, 367, 379, 127, 197, 439, 227, 163, 499, 101, 73, 263, 547, 281, 577, 293, 601, 607

Offset: 1

Joseph L. Pe, Jan 24 2014

The first p+1 (p prime) is 2+1=3, so 3 is the first term of the sequence. The next is 3+1=4=2*2, and the prime 2 appears next, so it is the second term of the sequence. The next p+1 = 5+1 = 6 gives no new prime factor; neither do 7+1 = 8 and 11+1 = 12. 13+1 = 14 = 2*7 gives the new prime factor 7, so 7 is the third term of the sequence.

If "p+1" is changed to "p-1" we get A112037. - N. J. A. Sloane, Jan 24 2014

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A112037.

DeleteDuplicates[ First /@ Flatten[FactorInteger[1 + Prime@Range@200], 1]] (* Giovanni Resta, Jan 24 2014 *)

A236387 Numbers n such that sigma(n) is an oblong number.

Original entry on oeis.org

5, 6, 11, 19, 20, 26, 28, 29, 30, 39, 40, 41, 46, 51, 55, 58, 71, 86, 89, 99, 104, 109, 114, 116, 117, 125, 131, 135, 158, 177, 181, 201, 202, 203, 209, 216, 226, 236, 239, 245, 271, 278, 306, 336, 340, 352, 377, 379, 398, 410, 411, 419, 428, 442, 447, 461

Offset: 1

Joseph L. Pe, Jan 24 2014

An oblong number (A002378) is of the form k(k+1) where k is a natural number.

The subsequence of prime terms is A002327 (primes of form n^2 - n - 1). - Michel Marcus, Jan 09 2015

			sigma(40) = 90 = 9*10, an oblong number; so 40 is a term of the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000203, A002327, A002378.

Select[Range[500], IntegerQ@ Sqrt[1+4*DivisorSigma[1, #]] &] (* Giovanni Resta, Jan 24 2014 *)

a(12)-a(56) from Giovanni Resta, Jan 24 2014

A236386 Numbers m such that phi(m) is an oblong number.

Original entry on oeis.org

3, 4, 6, 7, 9, 13, 14, 18, 21, 25, 26, 28, 31, 33, 36, 42, 43, 44, 49, 50, 62, 66, 73, 86, 87, 91, 95, 98, 111, 116, 117, 121, 135, 146, 148, 152, 157, 161, 169, 174, 182, 190, 201, 207, 211, 216, 222, 228, 234, 237, 241, 242, 252, 268, 270, 287, 289, 305

Offset: 1

Joseph L. Pe, Jan 24 2014

An oblong number (A002378) is of the form k*(k+1) where k is a natural number.
From Bernard Schott, Feb 27 2023: (Start)
Subsequence of primes is A002383 because in this case phi(k^2+k+1) = k*(k+1).
Subsequence of oblong numbers is A359847 where k and phi(k) are both oblong numbers. (End)

			phi(13) = 12 = 3*4, an oblong number; so 13 is a term of the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000010, A002378, A002383, A359847.

Similar, but where phi(m) is: A039770 (square), A039771 (cube), A078164 (biquadrate), A096503 (repdigit), A117296 (palindrome), A360944 (triangular).

filter := m -> issqr(1 + 4*phi(m)) : select(filter, [$(1 .. 700)]); # Bernard Schott, Feb 26 2023

Select[Range[500], IntegerQ@Sqrt[1 + 4*EulerPhi[#]] &] (* Giovanni Resta, Jan 24 2014 *)

isok(m) = my(t=eulerphi(m)); !(t%2) && ispolygonal(t/2, 3); \\ Michel Marcus, Feb 27 2023

from itertools import count, islice
from sympy.ntheory.primetest import is_square
from sympy import totient
def A236386_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:is_square((totient(n)<<2)+1), count(max(1,startvalue)))
A236386_list = list(islice(A236386_gen(),20)) # Chai Wah Wu, Feb 28 2023

a(16)-a(58) from Giovanni Resta, Jan 24 2014

A171183 Numbers n such that sigmawt(n) = sigmawt(n+1), where sigmawt(n) is the sum of the divisors of n weighted by divisor multiplicity in n.

Original entry on oeis.org

14, 957, 1334, 1634, 2402, 2685, 20145, 33998, 42818, 74918, 79826, 79833, 84134, 111506, 122073, 138237, 147454, 166934, 201597, 274533, 289454, 347738, 383594, 416577, 440013, 544334, 605985, 649154, 655005, 1060802, 1642154, 1674513

Offset: 1

Joseph L. Pe, Dec 05 2009

nonn

Ray Chandler, Table of n, a(n) for n=1..200

See A168512 for definition of divisor multiplicity.

divmult[d_, n_] := Module[{output, i}, If[d == 1, output = 1, If[d == n, output = 1, i = 0; While[Mod[n, d^(i + 1)] == 0, i = i + 1]; output = i]]; output]; dmt[n_] := Module[{divs, l}, divs = Divisors[n]; l = Length[divs]; Sum[divmult[divs[[i]], n]*divs[[i]], {i, 1, l}]]; l = {}; Do[If[dmt[i] == dmt[i + 1], l = Append[l, i]], {i, 1, 10^6}]; l

Extended by Ray Chandler, Dec 08 2009

cp's OEIS Frontend

User: Joseph L. Pe

A333079 The largest nontrivial divisor of n equals the sum of the other nontrivial divisors of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A238161 Greatest common divisor of the prime factors of n, each increased by 1.

Views

Author

Keywords

Examples

Programs

Mathematica

A238162 Least common multiple of the prime factors of n, each increased by 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A237282 The sum of the totatives of n is a perfect cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A236693 Numbers k such that 2^sigma(k) == 1 (mod k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A235989 sigma(n) is an additive inverse of n modulo phi(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A236388 Primes in order of first appearance among the prime factors of p+1 where p is a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A236387 Numbers n such that sigma(n) is an oblong number.

Views

Author

Keywords

Comments